Jaime Castro Pérez, José Ríos Montes: A topology and the frame attached to a set of primitive submodules, 139-156

Abstract:

For a multiplication $R$-module $M$ we define the primitive topology $\mathcal{T}$ on the set $Prt\left( M\right)$ of primitive submodules of $M$. We prove that if $R$ is a commutative ring and $M$ is a multiplication $R$-module, then the complete lattice $Sprt\left( M\right)$ of semiprimitive submodules of $M$ is a spatial frame. When $M$ is projective in the category $\sigma [M]$, we obtain that the topological spaces $(Prt(M),\mathcal{T}$) and $(Prt(R),\mathcal{T}$) are homeomorphic. As an application, we prove that if $M$ is projective in the category $\sigma [M]$, then $Prt(R)$ has classical Krull dimension if and only if $Prt(M)$ has classical Krull dimension.

Key Words: Multiplication module, primitive submodule, spatial frame, Krull dimension.

2020 Mathematics Subject Classification: Primary 16D60; Secondary 16P40, 16S90.

Download the paper in pdf format here.